Differential Analysis: Perturbation and Quasianalyticity

Perturbation is a fundamental concept in mathematics as well as the physical and applied sciences, which is based on the idea to study a system deviating slightly from an ideal system for which the solution is known. For instance, perturbation theory for linear operators, founded in the 1920s by Rayleigh and Schrödinger as part of studying acoustics and quantum mechanics, is concerned with the behavior of the spectral properties when the operator undergoes a small change. A central problem of this theory is the regularity of the roots of a parameterized family of polynomials (i.e. perturbation of polynomials). Due to its fundamental nature, perturbation of polynomials appears in areas as diverse as partial differential equations (PDEs) and algebraic geometry. Holmgren`s work on the heat equation at the beginning of the twentieth century showed that the solutions of certain PDEs live in so-called ultradifferentiable function spaces defined by bounds on the iterated derivatives. Ultradifferentiable functions form classes intermediate between real analytic and smooth functions which interpolate between the two extremes and thus play an important role in the regularity theory of PDEs. Ultradifferentiable classes are either quasianalytic, if each element is uniquely determined by its infinite Taylor expansion at any point, or non-quasianalytic. Quasianalyticity, discovered at the end of the nineteenth century by Émile Borel, generalizes the uniqueness properties of analytic functions and appears naturally in harmonic analysis and PDEs. Over the last decade, quasianalyticity became increasingly important in tame real geometry, partly due to the fact that resolution of singularities is available for quasianalytic classes (and on the other hand requires quasianalyticity). However, other standard techniques of local analytic geometry are missing, which makes quasianalytic analysis technically challenging. The main purpose of this project is to determine the optimal regularity of roots of polynomials as well as to study ultradifferentiable functions with particular emphasis on quasianalyticity in finite and infinite dimensions and to solve some of its classical open problems. These aims are intertwined by their common applications to perturbation theory and PDEs. The regularity problem for roots of polynomials may be seen as a particular case of a general lifting problem for representations of reductive algebraic or compact Lie groups. It thus constitutes one aspect of the broader objective of studying analytic and geometric properties of orbit spaces.

In many mathematical problems of most different nature one encounters the question, how regular the roots of a polynomial with regular coefficients can be chosen. This nonlinear problem was solved in the project: we were able to find the optimal regularity of the roots under minimal conditions. The uniformity of our estimates entails versatile applicability of the results. The new method, that we developed for the proofs, will contribute to the solution of a number of other problems in differential analysis and beyond. The closely related perturbation theory of linear operators, for example, has found an application in cosmology for boson stars within the framework of the project. Another focus of the project was the study of ultradifferentiable function classes. These are subclasses of the class of infinitely often differentiable functions, which play an important role in the theory of partial differential equations. There are two historical, not fully compatible approaches to these classes. Within the framework of the project we developed an ultradifferentiable theory that encompasses both approaches and we successfully used it to solve some open problems in the field. For example, a calculus for ultradifferentiable analysis in infinite dimensions was developed, extension properties of ultradifferentiable Whitney jets were characterized, and applications in microlocal analysis and asymptotic expansion and summability theory were given. Of particular interest in geometry are the quasianalytic ultradifferentiable classes, whose elements are already uniquely determined by the successive derivatives in one point. We were able to prove that quasianalytic ultradifferentiability (in contrast to non-quasianalytic ultradifferentiability) can never be tested in lower dimensions. A related topic extensively studied in the project were diffeomorphism groups of different regularity in n-dimensional Euclidean space. For a large number of regularity classes we could show that the respective diffeomorphisms form an infinitely dimensional Lie group. As an application we solved the 1-dimensional Hunter-Saxton equation, which occurs in the study of liquid crystals, in some of these classes. With regard to finite differentiability (in the Banach space setting) we studied groups of Hölder and Besov diffeomorphisms. These are not topological groups, but nevertheless, we were able to prove strong regularity properties, in particular closedness under the solution of ordinary differential equations. Progress has also been made in analysis on singular spaces. On the one hand lifting problems for orbit spaces of group actions were studied, on the other hand we found a large class of singular spaces for which smoothness of functions defined on these spaces can be tested on smooth curves.